400 PHILOSOPHICAL TRANSACTIONS. [ANNO iGSJ . 



pz -\- b = O, there were two affirmative roots and one negative root, or one 

 negative only ; viz. accordingly as -^q is either greater or less than y'cP + -^\b* 



If the third term, or pz, be wanting, the centre R always falls in the line 

 IPEA; wherefore, if it be z^ — bz^. ^. — q, or z^ -}- tz.'^ >fc. + q, there 

 can be but one root, affirmative, if it be — b, negative if + b. But if it be 

 z^ — "^z^. >|c. + 9, or z^ + bi^. 9fi. — q, there may be two affirmative roots 

 and one negative in the former^ or one affirmative and two negatives in the 

 latter, the centra falling in the line Pa between P and A, that is, if \q be 

 less than -^b^ ; but if it be greater, there can be only one negative root in the 

 former, or one affirmative in the latter. 



Hitherto we have fully considered the number of roots in cubic equations ; 

 it remains that we add something concerning the quantity of the roots. And 

 here it is first to be noted, that every equation, having three roots, may be 

 expeditiously resolved by means of the tables of sines, viz. by the trisection of 

 sn angle; that is, putting v^|-63 — ^p, or »/ Ad = the radius of the circle, if 

 it be + p in the equation, or *^aZ>^ 4- -^p, if — p; and the angle trisected, 



will have its sine in the tables of sines "^ ^ — — . This angle being 



found, the sine of its third part, as also the sine of the third part of its supple- 

 ment to a semicircle, their sum will be given f rom the tab le of sines, and these 

 sines are to be multiplied into the radius ^ ^bb + tPj and thus will be ob- 

 tained the quantities, 3/, y, y, in the figure ; the sum, or difl^erence of which and 

 -ji^b, as the case requires, will give the true roots of the equation. All these 

 things are deduced from Descartes's discoveries. But that all the cases may be 

 comprehended with as much brevity as possible ; the centre R, in the first for- 

 mula of equations, falling in the space VGP, the two intersections YY, fall 

 between A and B, and consequently either of the lesser roots is less than -^b, 

 but the third and greater always exceed -^6, but are exceeded by b : but if the 

 centre fall in the space GNV, there are two greater than -^b, but less than ^b; 

 and the third is 6 — the two others, and consequently less than -^b ; but using 

 the limitation of the quantity p, the roots are included in narrower bou nds ; for 

 the greatest root is less than '^ -^b"^ — f /^ + i^j but greate r than ^-^b b — p 

 -f- ^b; but when 4-66 is less than/?, that limit becomes ^ ^bb — -yp + 0. 

 The middle root is always less than ^ -^b"^ — p + -^b, but greater than -yb — 

 1/4.6^ — i-p ; and the least root never exceeds this limit, but vanishes with the 

 quantity q. 



In the second formula, according to the prescribed laws, there are two affir- 

 mative roots and one negative; and the centre falling in the space GPE, one 

 of the affirmative is greater and the other less than -^b, but the greater does 

 not exceed 6; and the negative cannot be greater than ^i^bb — ^b, being 



